03 July 2021
AperTO - Archivio Istituzionale Open Access dell'Università di Torino
Original Citation:
Sharp integral bounds for Wigner distributions
Published version:
DOI:10.1093/imrn/rnw250
Terms of use:
Open Access
(Article begins on next page)
Anyone can freely access the full text of works made available as "Open Access". Works made available under a Creative Commons license can be used according to the terms and conditions of said license. Use of all other works requires consent of the right holder (author or publisher) if not exempted from copyright protection by the applicable law.
Availability:
This is a pre print version of the following article:
arXiv:1605.00481v3 [math.FA] 27 Jun 2016
ELENA CORDERO AND FABIO NICOLA
Abstract. The cross-Wigner distribution W (f, g) of two functions or temper-ate distributions f, g is a fundamental tool in quantum mechanics and in signal analysis. Usually, in applications in time-frequency analysis f and g belong to some modulation space and it is important to know which modulation spaces W(f, g) belongs to. Although several particular sufficient conditions have been appeared in this connection, the general problem remains open. In the present paper we solve completely this issue by providing the full range of modulation spaces in which the continuity of the cross-Wigner distribution W (f, g) holds, as a function of f, g. The case of weighted modulation spaces is also considered. The consequences of our results are manifold: new bounds for the short-time Fourier transform and the ambiguity function, boundedness results for pseudodifferential (in particular, localization) operators and properties of the Cohen class.
1. Introduction
The (cross-)Wigner distribution was first introduced in physics to account for quantum corrections to classical statistical mechanics in 1932 by Wigner [50] and in 1948 it was proposed in signal analysis by Ville [49]. This is why the Wigner distribution is also called Wigner-Ville distribution. Nowadays it can be considered one of the most important time-frequency representations, second only to the spec-trogram, and it is one of the most commonly used quasiprobability distribution in quantum mechanics [21, 28].
Given two functions f1, f2 ∈ L2(Rd), the cross-Wigner distribution W (f1, f1) is defined to be (1) W (f1, f2)(x, ξ) = Z f1(x + t 2)f2(x − t 2)e −2πiξtdt.
The quadratic expression W f = W (f, f ) is called the Wigner distribution of f . An important issue related to such a distribution is the continuity of the map (f1, f2) 7→ W (f1, f2) in the relevant Banach spaces. The basic result in this con-nection is the easily verified equality
kW (f1, f2)kL2(R2d) = kf1kL2(Rd)kf2kL2(Rd). 2010 Mathematics Subject Classification. 42B10,81S30,42B37,35S05.
Key words and phrases. Wigner distribution, time-frequency representations, modulation spaces, Wiener amalgam spaces.
Beside L2, the time-frequency concentration of signals is often measured by the so-called modulation space norm Mp,q
m , 1 ≤ p, q ≤ ∞, for a suitable weight function m (cf. [22, 23, 28] and Section 2 below). In short, these spaces are defined as follows. For a fixed non-zero g ∈ S(Rd), the short-time Fourier transform (STFT) of f ∈ S′(Rd) with respect to the window g is given by
(2) Vgf (x, ξ) =
Z
Rd
f (t) g(t− x) e−2πiξtdt . Then the space Mp,q
m (Rd) is defined by Mp,q
m (Rd) = {f ∈ S′(Rd) : Vgf ∈ Lp,qm (R2d)} endowed with the obvious norm. Here Lp,q
m (R2d) are mixed-norm weighted Lebesgue spaces in R2d; see Section 2 below for precise definitions.
Both the STFT Vgf and the cross-Wigner distribution W (f, g) are defined on many pairs of Banach spaces. For example, they both map L2(Rd) × L2(Rd) into L2(R2d) and S(Rd)×S(Rd) into S(R2d) and can be extended to a map from S′(Rd)× S′(Rd) into S′(R2d).
In this paper we will mainly work with the polynomial weights (3) vs(z) = hzis= (1 + |z|2)
s
2, z ∈ R2d, s ∈ R.
For w = (z, ζ) ∈ R4d, we write (1 ⊗ vs)(w) = vs(ζ). Now, the problem addressed in this paper is to provide the full range of exponents p1, p2, q1, q2, p, q ∈ [1, ∞] such that
kW (f1, f2)kM1⊗vsp,q .kf1kMvsp1,q1kf2kMvsp2,q2.
These estimates were proved in [48, Theorem 4.2] (cf. also [47, Theorem 4.1] for modulation spaces without weights) under the conditions
p ≤ pi, qi ≤ q, i = 1, 2 and (4) 1 p1 + 1 p2 = 1 q1 + 1 q2 = 1 p + 1 q.
However, it is not clear whether these conditions are necessary as well.
Our main result shows that the sufficient conditions can be widened and such extension is sharp.
Theorem 1.1. Assume pi, qi, p, q ∈ [1, ∞], s ∈ R, such that
(5) pi, qi ≤ q, i = 1, 2 and that (6) 1 p1 + 1 p2 ≥ 1 p+ 1 q, 1 q1 + 1 q2 ≥ 1 p + 1 q.
Then, if f1 ∈ Mvp1,q1|s| (R d) and f 2 ∈ Mvsp2,q2(Rd) we have W (f1, f2) ∈ M1⊗vp,qs(R 2d), and (7) kW (f1, f2)kM1⊗vsp,q .kf1kMv|s|p1,q1kf2kMvsp2,q2. Viceversa, assume that there exists a constant C > 0 such that (8) kW (f1, f2)kMp,q ≤ Ckf1kMp1,q1kf2kMp2,q2, ∀f1, f2 ∈ S(R2d). Then (5) and (6) must hold.
The remarkable fact of this result, in our opinion, is that the conditions (5) and (6) turn out to be necessary too.
The consequences of this are manifold. First, in the framework of signal analysis and time-frequency representations, we obtain new estimates for the short-time Fourier transform Vf1f2 and the ambiguity function A(f1, f2) (see Section 2 for definitions). In particular, we recapture the sharp Lieb’s bounds in [36, Theorem 1]
kA(f1, f2)kLq .kf1kL2kf2kL2,
valid for q ≥ 2; we also refer to [14, 15] for related estimates for the short-time Fourier transform and [11] for the strictly related Born-Jordan distribution.
Secondly, we easily provide new boundedness results for pseudodifferential op-erators (in particular, localization opop-erators) with symbols in modulation spaces. Let us mention that the study of pseudodifferential operators in the context of modulation spaces has been pursued by many authors. The earliest works are due to Sj¨ostrand [42] and Tachizawa [45]. In the former work pseudodifferen-tial operators with symbols in the modulation space M∞,1 (also called Sj¨ostrand’s class) where investigated. Later, sufficient and some necessary boundedness con-ditions where investigate by Gr¨ochenig and Heil [29, 30] and Labate [34, 35]. Since the year 2003 until today the contributions on this topic are so multi-plied that there is hard to mention them all. Let us just recall some of them [2, 4, 18, 19, 31, 33, 38, 39, 43, 44, 47, 48].
Every continuous operator from S(Rd) to S′(Rd) can be represented as a pseudo-differential operator in the Weyl form Lσ and the connection with the cross-Wigner distribution is provided by
(9) hLσf, gi = hσ, W (g, f)i, f, g ∈ S(Rd).
By using this formula we can translate the boundedness results for the cross-Wigner distribution in Theorem 1.1 to boundedness results for Weyl operators.
Pseudodifferential operators of great interest in signal analysis are the so-called localization operators Aϕ1,ϕ2
a (see Section 5), which can be represented as Weyl operators as follows (cf. [6, 13, 46])
so that the Weyl symbol of the localization operator Aϕ1,ϕ2
a is given by (11) σ = a ∗ W (ϕ2, ϕ1) .
Using this representation of localization operators as Weyl operators and using The-orem 1.1 we are able to obtain new boundedness results for localization operators, see Theorem 5.2 in Section 5 below.
Finally, another application of Theorem 1.1 is the investigation of the time-frequency properties of the Cohen class, introduced by Cohen in [8]. This class consists of elements of the type
(12) M(f, f ) = W (f, f ) ∗ σ
where σ ∈ S′(R2d) is called the Cohen kernel. When σ = δ, then M(f, f ) = W (f, f ) and we come back to the Wigner distribution. For other choices of kernels we recapture the Born-Jordan distribution [10, 11, 12] or the τ -Wigner distributions Wτ(f, f ) [7, Proposition 5.6]. In this framework we have the following result. Theorem 1.2. Assume s ≥ 0, p1, q1, p, q ∈ [1, ∞] such that
(13) 2 min{1 p1 , 1 q1} ≥ 1 p + 1 q.
Consider a Cohen kernel σ ∈ M1,∞(R2d). If f ∈ Mp1,q1
vs (Rd), then the Cohen distribution M(f, f ) is in M1⊗vp,qs(R2d), with
(14) kM(f, f)kM1⊗vsp,q (R2d) .kσkM1,∞(R2d)kfk2
Mvsp1,q1(Rd).
In particular, the τ -kernels and the Born-Jordan kernels enjoy such a property, cf. Section 6.
For the sake of clarity our results have been presented only for the polynomial weights vs, but we remark that more general weights can also be considered, see the following Remark 3.2, (i).
Further developments of this research could involve the study of boundedness for bilinear/multilinear pseudodifferential operators and localization operators. This requires an extension of Theorem 1.1 to more general Wigner/Rihaczek distributi-ons and STFT, see e.g., [3, 16, 27] and the recent contribution [37]. We leave this study to a subsequent paper.
In short, the paper is organized as follows. Section 2 is devoted to some pre-liminary results from time-frequency analysis and in particular to the computation of the STFT of a generalized Gaussian and its modulation norm. In Section 3 we prove Theorem 1.1. In Section 4 we show the continuity properties of pseudodiffe-rential (and in particular localization) operators on modulation spaces. In Section 5 we present a time-frequency analysis of the Cohen class.
Notation. We define t2 = t · t, for t ∈ Rd, and xy = x · y is the scalar product on Rd. The Schwartz class is denoted by S(Rd), the space of tempered distributions by S′(Rd). We use the brackets hf, gi to denote the extension to S′(Rd) × S(Rd) of the inner product hf, gi = R f (t)g(t)dt on L2(Rd). The Fourier transform is normalized to be ˆf (ξ) = Ff(ξ) =R f(t)e−2πitξdt, the involution g∗ is g∗(t) = g(−t). The operators of translation and modulation are defined by Txf (t) = f (t − x) and Mξf (t) = e2πitξf (t), x, ξ ∈ Rd.
2. Preliminaries
2.1. Modulation spaces. The modulation and Wiener amalgam space norms are a measure of the joint time-frequency distribution of f ∈ S′. For their basic properties we refer to the original literature [22, 23, 24] and the textbooks [21, 28]. For the description of the decay properties of a function/distribution, weight functions on the time-frequency plane are employed. We denote by v a continu-ous, positive, even, submultiplicative weight function (in short, a submultiplica-tive weight), i.e., v(0) = 1, v(z) = v(−z), and v(z1 + z2) ≤ v(z1)v(z2), for all z, z1, z2 ∈ R2d. A positive, even weight function m on R2d is called v-moderate if m(z1 + z2) ≤ Cv(z1)m(z2) for all z1, z2 ∈ R2d. Observe that vs is a v|s|-moderate weight, for every s ∈ R. Given a non-zero window g ∈ S(Rd), a v-moderate weight function m on R2d, 1 ≤ p, q ≤ ∞, the modulation space Mp,q
m (Rd) consists of all tempered distributions f ∈ S′(Rd) such that the STFT V
gf (defined in (2)) is in Lp,q
m (R2d) (weighted mixed-norm spaces), with norm
kfkMmp,q = kVgf kLp,qm = Z Rd Z Rd|V gf (x, ξ)|pm(x, ξ)pdx q/p dξ !1/q . (Obvious modifications occur when p = ∞ or q = ∞). If p = q, we write Mp
m instead of Mp,p
m , and if m(z) ≡ 1 on R2d, then we write Mp,q and Mp for Mmp,q and Mp,p
m . Then Mmp,q(Rd) is a Banach space whose definition is independent of the choice of the window g, in the sense that different nonzero window functions yield equivalent norms. The modulation space M∞,1 is also called Sj¨ostrand’s class [42]. We now recall the definition of the Wiener amalgam spaces that are image of the modulation spaces under the Fourier transform. For any even weigh functions u, w on Rd, the Wiener amalgam spaces W (FLp
u, Lqw)(Rd) are given by the distributions f ∈ S′(Rd) such that kfkW (FLpu,Lqw)(Rd) := Z Rd Z Rd|Vgf (x, ξ)| pup(ξ) dξ q/p wq(x)dx !1/q < ∞ (with natural changes for p = ∞ or q = ∞). Using Parseval identity we can write the so-called fundamental identity of time-frequency analysis Vgf (x, ξ) =
e−2πixξV ˆ
gf (ξ, −x), hence |Vˆ gf (x, ξ)| = |Vˆgf (ξ, −x)| so that (recall u(x) = u(−x))ˆ (15) kfkMu⊗wp,q = k ˆf kW (FLpu,Lqw).
This proves that these Wiener amalgam spaces are the image under Fourier trans-form of modulation spaces:
(16) F(Mu⊗wp,q ) = W (FLpu, Lqw).
In the sequel we will need the inclusion relations for modulation spaces. Assume m1, m2 ∈ Mv(R2d), then
(17) S(R
d
) ⊆ Mm1p1,q1(Rd) ⊆ Mm2p2,q2(Rd) ⊆ S′(Rd), if p1 ≤ p2, q1 ≤ q2, m2 .m1.
Moreover, we will often apply convolution relations for modulation spaces [13, Proposition 2.1] for the vs weight functions as follows.
Proposition 2.1. Let ν(ξ) > 0 be an arbitrary weight function on Rd, s ∈ R, and 1 ≤ p, q, u, v, t ≤ ∞. If 1 p+ 1 q − 1 = 1 u, and 1 t + 1 t′ = 1 v , then (18) M1⊗νp,t (Rd) ∗ M1⊗v|s|q,t′ ν−1(R d ) ֒→ Mvsu,v(Rd) with norm inequality kf ∗ hkMvsu,v .kfkM1⊗νp,t khkMq,t′
1⊗v|s|ν−1.
2.2. Time-frequency tools. To prove our main result, we will need to compute the STFT of the cross-Wigner distribution, proved in [28, Lemma 14.5.1]:
Lemma 2.1. Fix a nonzero g ∈ S(Rd) and let Φ = W (g, g) ∈ S(R2d). Then the STFT of W (f1, f2) with respect to the window Φ is given by
(19) VΦ(W (f1, f2))(z, ζ) = e−2πiz2ζ2Vgf2(z1+ ζ2 2, z2− ζ1 2)Vgf1(z1− ζ2 2, z2+ ζ1 2) . Lemma 2.2. Consider the Gaussian function ϕ(x) = e−πx2
and its rescaled ver-sion ϕλ(x) = e−πλx
2
, λ > 0. Then the cross-Wigner distribution is the following Gaussian function (20) W (ϕ, ϕλ)(x, ξ) = 2d (1 + λ)d2 e−aλπx2e−bλπξ2e2πicλxξ with (21) aλ = 4λ 1 + λ bλ = 4 1 + λ cλ = 2(1 − λ) 1 + λ .
Proof. The proof is obtained by an easy computation. In particular, we will make the change of variables t = √2s
1+λ − 2√1−λ1+λx, so that dt = 2d (1+λ)d/2ds. In details, W (ϕ, ϕλ)(x, ξ) = Z Rd e−π x+2t 2 −πλ x−t2 2 e−2πitξ dt = e−π(1+λ)x2 Z Rd e−π4 (1+λ)t2+4(1−λ)xt e−2πitξ dt = e−π(1+λ)x2 Z Rd e−π4 √ 1+λt+2(1−λ)√ 1+λx 2 eπ(1−λ)21+λ x2e−2πitξ dt = e−π (1+λ)−(1−λ)21+λ x2Z Rd e−π4 √ 1+λt+2√1−λ 1+λx 2 e−2πitξ dt = e−π1+λ4λ x2 Z Rd e−πs2 e−2πi √1+λ2s − 2(1−λ) 1+λ x ξ 2d (1 + λ)d/2 ds = 2 d (1 + λ)d/2e −π1+λ4λ x2e4πi1−λ1+λxξ Z Rd e−πs2e−2πis √2ξ1+λ ds = 2 d (1 + λ)d/2e −π1+λ4λ x2e−π 4 1+λξ2e4πi1−λ1+λxξ, as desired.
Hence the Wigner distribution above is a generalized Gaussian. Our goal will be to compute the modulation norm of this Wigner distribution. The first step is the calculation of the STFT of a generalized Gaussian.
Proposition 2.2. Given a, b, c > 0, consider the generalized Gaussian function (22) f (x, ξ) = e−πax2e−πbξ2e2πicxξ, (x, ξ) ∈ R2d. For Φ(x, ξ) = e−π(x2+ξ2) , z = (z1, z2), ζ = (ζ1, ζ2) ∈ R2d, we obtain VΦf (z, ζ) = 1 [(a + 1)(b + 1) + c2]d2e
−π[a(b+1)+c2]z21 +[(a+1)b+c2]z22 +(b+1)ζ1 +(a+1)ζ2 2 −2c(z1ζ2+z2ζ1 )2 (a+1)(b+1)+c2
× e−a+12πi
z1ζ1+(cz1−(a+1)ζ2)(a+1)(b+1)+c2cζ1+(a+1)z2 . (23)
Proof. We write VΦf (z, ζ) =
Z
R2d
e−πax2−πbξ2+2πicxξe−2πi(ζ1x+ζ2ξ)e−π[(x−z1)2+(ξ−z2)2]dxdξ = e−π(z12+z22) Z Rd Z Rd e−π[(a+1)x2−2xz1]e−2πix(ζ1−cξ)dx × e−π[(b+1)ξ2−2ξz2]e−2πiζ2ξ dξ = e−π 1−a+11 z21−πz2 2 Z Rd Z Rd e−π √ a+1x−√z1a+1 2 e−2πix(ζ1−cξ)dx × e−π[(b+1)ξ2−2ξz2]e−2πiζ2ξ dξ. With the change of variables√a + 1x − √z1
a+1 = t, dx = dt (a+1)d/2, we obtain VΦf (z, ζ) = 1 (a + 1)d/2e −π a a+1z21−πz22 Z Rd Z Rd
e−πt2e−2πi √a+1t +a+1z1
(ζ1−cξ)dt × e−π[(b+1)ξ2−2ξz2]e−2πiζ2ξ dξ
= 1
(a + 1)d/2e
−πa+1a z21−πz22−2πiz1ζ1a+1 Z
Rd
e−π(ζ1−cξ) 2
a+1 e2πia+1cz1ξ
× e−π[(b+1)ξ2−2ξz2]e−2πiζ2ξ dξ
= 1
(a + 1)d/2e −π a
a+1z21−πz22−2πiz1ζ1a+1−π ζ21 a+1 × Z Rd e−π (b+1)ξ2+ c2 a+1ξ2−2a+1cζ1ξ−2ξz2 e2πi a+1cz1−ζ2 ξ dξ. (24)
The last integral can be computed as follows: I :=
Z
Rd e−π
(a+1)(b+1)+c2
a+1 ξ2−2cζ1+z2(a+1)a+1 ξ e2πi a+1cz1−ζ2 ξ dξ = Z Rd
e−a+1π {[(a+1)(b+1)+c2]ξ2−2[cζ1+z2(a+1)]ξ}e2πi a+1cz1−ζ2 ξ dξ = Z Rd e− π a+1 h√ (a+1)(b+1)+c2ξ−√cζ1+z2(a+1) (a+1)(b+1)+c2 i2 ea+1π [cζ1+z2(a+1)] 2
(a+1)(b+1)+c2e2πi a+1cz1−ζ2 ξ dξ = ea+1π [c2ζ21 +(a+1)2] (a+1)(b+1)+c2 Z Rd e− π a+1 h√ (a+1)(b+1)+c2ξ−√cζ1+z2(a+1) (a+1)(b+1)+c2 i2 e2πi a+1cz1−ζ2 ξ dξ.
Making the change of variables t = √ (a+1)(b+1)+c2ξ−√cζ1+z2(a+1) (a+1)(b+1)+c2 √ a+1 , so that dξ = a+1 (a+1)(b+1)+c2 d/2 dt, we can write I = (a + 1) d 2 [(a + 1)(b + 1) + c2]d2 Z Rd e−πt2e2πi cz1 a+1−ζ2 h √a+1 √ (a+1)(b+1)+c2t− cζ1+(a+1)z2 (a+1)(b+1)+c2 i dt = (a + 1) d 2 [(a + 1)(b + 1) + c2]d2 ea+1π (a+1)2z22 +c2ζ21 +2c(a+1)z2ζ1 (a+1)(b+1)+c2 e−2πi cz1 a+1−ζ2 cζ1+(a+1)z2 (a+1)(b+1)+c2 × e−π h a+1 (a+1)(b+1)+c2 cz1 a+1−ζ2 2i .
The result then follows by substituting the value of the integral I in (24).
Corollary 2.3. Consider the generalized Gaussian f defined in (22) and the win-dow function Φ(x, ξ) = e−π(x2+ξ2)
. Then, for every 1 ≤ p, q ≤ ∞, we have (25) kfkMp,q ≍ kVΦf kLp,q ≍ [(a + 1)(b + 1) + c 2]dp+ d q− d 2[(c2+ ab + a)(c2+ ab + b)] d 2q− d 2p [b2(a + 1) + b(c2+ a + 1)]2qd[a2(b + 1) + a(c2+ b + 1)] d 2q . The cases p = ∞ or q = ∞ can be obtained by using the rule 1/∞ = 0 in formula (25).
Proof. By Proposition 2.2, we can write |VΦf (z, ζ)| =
1
[(a + 1)(b + 1) + c2]d2 e−π
[a(b+1)+c2]z21 +[(a+1)b+c2]z22 +(b+1)ζ21 +(a+1)ζ22 −2c(z1ζ2 +z2ζ1)
(a+1)(b+1)+c2 .
It remains to compute the mixed Lp,q-norm of the previous function. We treat the cases 1 ≤ p, q < ∞. The cases either p = ∞ or q = ∞ are obtained with obvious modifications.
For simplicity, we set
α = c 2 + a(b + 1) (a + 1)(b + 1) + c2, β = c2+ (a + 1)b (a + 1)(b + 1) + c2, γ = (b + 1) (a + 1)(b + 1) + c2, (26) δ = (a + 1) (a + 1)(b + 1) + c2, σ = c (a + 1)(b + 1) + c2. (27) Hence (28) kVΦf kLp,q [(a + 1)(b + 1) + c2]−2pd = Z R2d Ipqe−πq(γζ12+δζ22) dζ 1dζ2 1q =: A,
where I := R
R2de−πpαz 2
1−πpβz22e2πpσ(z1ζ2+z2ζ1) dz1dz2. Now straightforward computa-tions and change of variables yield
I = Z Rd Z Rd e−πp(αz21−2σζ2z1) dz 1 e−πβz22+2πpσz2ζ1 dz 2 = eπpσ2αζ22eπp σ2 β ζ12 Z Rd e−πp(√αz1−√σαζ2)2 dz 1 Z Rd e−πp(√βz2−√σβζ1) 2 dz2 = p−d2α−d2p−d2β−d2eπpσ2αζ22eπp σ2 βζ12.
Substituting the value of the integral I in (28), we obtain A = p−dpα− d 2pβ− d 2p Z Rd eπqσ2β ζ12−πqγζ12 dζ 1 Z Rd eπqσ2αζ22−πqδζ22 dζ 2 1q = p−dpα−2pdβ−2pd Z Rd e−πq(γ−σ2β )ζ12 dζ 1 Z Rd e−πq(δ−σ2α)ζ22 1q = p−dpα−2pdβ−2pdq−dq γ − σ 2 β −2qd γ − σ 2 α −2qd .
Finally, the goal is attained by substituting in A the values of the parameters α, β, γ, δ, σ in (26) and (27) and observing that
kfkMp,q ≍ kVΦf kLp,q = A[(a + 1)(b + 1) + c2]− d 2. This concludes the proof.
We have now all the tools to compute the modulation norm of the (cross-)Wigner distribution W (ϕ, ϕλ) in (20). Precisely, setting in formula (25) the values a = aλ, b = bλ, c = cλ, where aλ, bλ, cλ are defined in (21), and making easy simplifications we attain the following result.
Corollary 2.4. For λ > 0 consider the (cross-)Wigner distribution W (ϕ, ϕλ) de-fined in (20) (cf. Lemma 2.2). Then
(29) kW (ϕ, ϕλ)kMp,q ≍ [(2λ + 1)(λ + 2)] d 2q−2pd λ2qd(1 + λ) d 2− d p .
The cases p = ∞ or q = ∞ can be obtained by using the rule 1/∞ = 0 in formula (29).
3. Main Result
In this Section we prove Theorem 1.1. We will focus separately on the sufficient and necessary part in the statement.
Theorem 3.1 (Sufficient Conditions). If p1, q1, p2, q2, p, q ∈ [1, ∞] are indices which satisfy (5) and (6), s ∈ R, f1 ∈ Mvp1|s|,q1(Rd) and f2 ∈ Mvsp2,q2(Rd), then W (f1, f2) ∈ M1⊗vsp,q (R2d), and the estimate (7) holds true.
Proof. We first study the case p, q < ∞. Let g ∈ S(Rd) and set Φ = W (g, g) ∈ S(R2d). If ζ = (ζ
1, ζ2) ∈ R2d, we write ˜ζ = (ζ2, −ζ1). Then, from Lemma 2.1, (30) |VΦ(W (f1, f2))(z, ζ)| = |Vgf2(z + ˜ ζ 2)| |Vgf1(z − ˜ ζ 2)| . Consequently, kW (f1, f2)kM1⊗vsp,q ≍ Z R2d Z R2d|V gf2(z + ˜ ζ 2)| p |Vgf1(z − ˜ ζ 2)| pdz qp hζisqdζ !1/q . After the change of variables z 7→ z − ˜ζ/2, the integral over z becomes the convolu-tion (|Vgf2|p∗|(Vgf1)∗|p)(˜ζ), and observing that (1⊗vs)(z, ζ) = hζis = vs(ζ) = vs(˜ζ), we obtain kW (f1, f2)kM1⊗vsp,q ≍ Z Z R2d(|Vg f2|p∗ |(Vgf1)∗|p) q p(˜ζ)v s(˜ζ)qdζ 1/p = k |Vgf2|p ∗ |(Vgf1)∗|pk 1 p L q p vps . Hence (31) kW (f1, f2)kpMp,q 1⊗vs ≍ k |Vgf2| p ∗ |(Vgf1)∗|pk L q p vps . Case p ≤ q < ∞.
Step 1. Here we prove the desired result in the case p ≤ pi, qi, i = 1, 2.
Suppose first that (4) are satisfied (and hence pi, qi ≤ q, i = 1, 2). Since q/p ≥ 1, we can apply Young’s Inequality for mixed-normed spaces (cf. [1], see also [26]) and majorize (31) as follows
kW (f1, f2)kpMp,q 1⊗vs .k |Vgf2| p kLr2,s2vp|s|k |(Vgf1)∗|pkLr1,s1vps = k|Vgf1|pkLr1,s1vp|s|k |Vgf2|pkLr2,s2vps = kVgf1kpLpr1,ps1 v|s| kVgf2k p Lpr2,ps2vs , for every 1 ≤ r1, r2, s1, s2 ≤ ∞ such that
(32) 1 r1 + 1 r2 = 1 s1 + 1 s2 = 1 + p q.
Choosing ri = pi/p ≥ 1, si = qi/p ≥ 1, i = 1, 2, the indices’ relation (32) becomes (4) and we obtain
Now, still assume p ≤ pi, qi, i = 1, 2 but 1 p1 + 1 p2 ≥ 1 p + 1 q, 1 q1 + 1 q2 = 1 p + 1 q,
(hence pi, qi ≤ q, i = 1, 2). We set u1 = tp1, and look for t ≥ 1 (hence u1 ≥ p1) such that 1 u1 + 1 p2 = 1 p+ 1 q that gives 0 < 1 t = p1 p + p1 q − p1 p2 ≤ 1
because p1(1/p + 1/q) − p1/p2 ≤ p1(1/p1 + 1/p2) − p1/p2 = 1 whereas the lower bound of the previous estimate follows by 1/(tp1) = 1/p + 1/q − 1/p2 > 0 since p ≤ p2. Hence the previous part of the proof gives
kW (f1, f2)kM1⊗vsp,q .kf1kMv|s|u1,q1kf2kMvsp2,q2 .kf1kMv|s|p1,q1kf2kMvsp2,q2.
where the last inequality follows by inclusion relations for modulations spaces Mp1,q1
vs (Rd) ⊆ Mvsu1,q1(Rd) for p1 ≤ u1. The general case
1 p1 + 1 p2 ≥ 1 p + 1 q, 1 q1 + 1 q2 ≥ 1 p + 1 q, can be treated analogously.
Step 2. Assume now that pi, qi ≤ q, i = 1, 2, and satisfy relation (6). If at least one out of the indices p1, p2 is less than p, assume for instance p1 ≤ p, whereas p ≤ q1, q2, then we proceed as follows. We choose u1 = p, u2 = q, and deduce by the results in Step 1 (with p1 = u1 and p2 = u2) that
kW (f1, f2)kM1⊗vsp,q .kf1kMv|s|u1,q1kf2kMvsu2,q2 .kf1kMv|s|p1,q1kf2kMvsp2,q2
where the last inequality follows by inclusion relations for modulation spaces, since p1 ≤ u1 = p and p2 ≤ u2 = q.
Similarly we argue when at least one out of the indices q1, q2 is less than p and p ≤ p1, p2 or when at least one out of the indices q1, q2 is less than p and at least one out of the indices q1, q2 is less than p. The remaining case p ≤ pi, qi ≤ q is treated in Step 1.
Case p = q = ∞. We use (30) and the submultiplicative property of the weight vs, kW (f1, f2)kM∞ 1⊗vs = sup z,ζ∈R2d|Vg f2(z + ˜ ζ 2)| |Vgf1(z − ˜ ζ 2)|vs(ζ) = sup z,ζ∈R2d||V gf2(z)| |(Vgf1)∗(z − ˜ζ)|vs(ζ) = sup z,ζ∈R2d||Vg f2(z)| |(Vgf1)∗(z − ˜ζ)|vs(˜ζ) ≤ sup z∈R2d(kV gf1v|s|k∞|Vgf2(z)vs(z)|) = kVgf1v|s|k∞kVgf2vsk∞ ≍ kfkM∞ v|s|kgkMvs∞ ≤ kfkMv|s|p1,q1kfkMvsp2,q2,
for every 1 ≤ pi, qi ≤ ∞, i = 1, 2. Notice that in this case conditions (5) and (8) are trivially satisfied.
Case p > q. Using the inclusion relations for modulation spaces, we majorize kW (f1, f2)kM1⊗vsp,q .kW (f1, f2)kM1⊗vsq,q .kf1kMv|s|p1,q1kf2kMvsp2,q2
for every 1 ≤ pi, qi ≤ q, i = 1, 2. Here we have applied the case p ≤ q with p = q. Notice that in this case condition (8) is trivially satisfied, since from p1, qi ≤ q we infer 1/p1+ 1/p2 ≥ 1/q + 1/q, 1/q1+ 1/q2 ≥ 1/q + 1/q. This concludes the proof. Remark 3.2. (i) The result of Theorem 3.1 can be extended to more general weights. In particular, it holds for polynomial weights satisfying relation (4.10) in [48]. Hence our result extends Toft’s result [48, Theorem 4.2] (cf. also [47, Theorem 4.1] for modulation spaces without weights). Other examples of suitable weights are given by sub-exponential weights of the type v(z) = eα|z|β
for α > 0 and 0 < β < 1.
(ii) The particular case p = 1, 1 ≤ q ≤ ∞, p1 = q1 = 1, p2 = q2 = q, s ≥ 0, was already proved in [13, Prop. 2.2].
(iii) For pi = qi = p = q = 2, i = 1, 2, we obtain the following continuity result for the cross-Wigner distribution acting between Shubin spaces and Sobolev spaces: For s ≥ 0, f1, f2 ∈ Qs(Rd) (cf. Shubin’s book [41]), the cross-Wigner distribution W (f1, f2) is in Hs(R2d) with
kW (f1, f2)kHs(R2d) .kf1kQs(Rd)kf2kQs(Rd).
(iv) Continuity properties of the cross-Wigner distribution on modulation spaces with different weight functions can be easily inferred using the techniques of The-orem 3.1 and the Young type inequalities for weighted spaces shown by Johansson et al. in [32, Theorem 2.2].
The estimate in (7) can be slightly improved if s ≥ 0. Precisely, we have the following result.
Theorem 3.3. If p1, q1, p2, q2, p, q ∈ [1, ∞] are indices which satisfy (5) and (6), s ≥ 0, f1 ∈ Mvsp1,q1(Rd) and f2 ∈ Mvsp2,q2(Rd), then W (f1, f2) ∈ M1⊗vsp,q (R2d), with
kW (f1, f2)kM1⊗vsp,q .kf1kMp1,q1kf2kMvsp2,q2 + kf1kMvsp1,q1kf2kMp2,q2.
Proof. The proof is similar to that of Theorem 3.1, but in this case from the estimate (31) we proceed by using
vs(z) . vs(z − w) + vs(w), s ≥ 0 (with sp in place of s) instead of vs(z) . v|s|(z − w)vs(w).
If in particular we consider the Wigner distribution W (f, f ), then Theorem 3.3 can be rephrased as follows.
Corollary 3.4. Assume s ≥ 0, p1, q1, p, q ∈ [1, ∞] such that 2 min{p1 1 , 1 q1} ≥ 1 p + 1 q. If f ∈ Mp1,q1
vs (Rd), then the Wigner distribution W (f, f ) is in M p,q
1⊗vs(R2d), with (33) kW (f, f)kM1⊗vsp,q (R2d).kfkMp1,q1(Rd)kfkMp1,q1
vs (Rd).
We now prove the sharpness of Theorem 3.1 (and Corollary 3.4) in the un-weighted case s = 0.
Theorem 3.5 (Necessary Conditions). Consider p1, p2, q1, q2, p, q ∈ [1, ∞]. As-sume that there exists a constant C > 0 such that
(34) kW (f1, f2)kMp,q ≤ Ckf1kMp1,q1kf2kMp2,q2, ∀f1, f2 ∈ S(R2d), then (5) and (6) must hold.
Proof. Let us first demonstrate the necessity of (6). We consider the dilated Gaus-sians ϕλ(x) = ϕ(
√
λx), with ϕ(x) = e−πx2 .
An easy computation (see also [28, formula (4.20)] (6)) shows that (35) W (ϕλ, ϕλ)(x, ξ) = 2
d 2λ−
d
2ϕ2λ(x)ϕ2/λ(ξ). Now (see [15, Lemma 3.2], [47, Lemma 1.8])
kϕλkMr,s ≍ λ− d 2r(λ + 1)− d 2(1− 1 q− 1 p) and observe that
kW (ϕλ, ϕλ)kMp,q(R2d) = 2 d
2λ−d2kϕ
2λkMp,q(Rd)kϕ2/λkMp,q(Rd). The assumption (8) in this case becomes
λ−d2(λ+1)− d
2(1−1q−1p)(λ−1+1)−d2(1−1q−p1) .λ− d
and letting λ → +∞ we obtain 1 p + 1 q ≤ 1 q1 + 1 q2 whereas for λ → 0+ 1 p+ 1 q ≤ 1 p1 + 1 p2 , so that (6) must hold.
It remains to prove the sharpness of (5). We first show the conditions p2, q2 ≤ q. We test (34) on the (cross-)Wigner distribution W (ϕ, ϕλ) defined in (20), that is
kW (ϕ, ϕλ)kMp,q(R2d) .kϕkMp1,q1(Rd)kϕλkMp2,q2(Rd). Using Corollary 2.4 the previous estimate can be rephrased as
[(2λ + 1)(λ + 2)]2qd− d 2p λ2qd(1 + λ)d2−dp .λ−2p2d λ− d 2(1− 1 q2− 1 p2), ∀λ > 0. Letting λ → +∞ we attain q2 ≤ q whereas for λ → 0+ p2 ≤ q.
The conditions p1, q1 ≤ q then follows by using the cross-Wigner property W (ϕλ, ϕ)(x, ξ) = W (ϕ, ϕλ)(x, ξ),
so that
kW (ϕλ, ϕ)kMp,q(R2d) = kW (ϕ, ϕλ)kMp,q(R2d) = kW (ϕ, ϕλ)kMp,q(R2d) and applying the same argument as before.
4. Continuity results for the short-time Fourier transform and the ambiguity distribution
This optimal bounds in Theorem 1.1 for the Wigner distribution can be trans-lated in optimal new estimates for other time-frequency representations such that the STFT or the ambiguity function. Precisely, given f, g ∈ L2(Rd), we recall the definition of the (cross-)ambiguity function
(36) A(f1, f2)(x, ξ) = Z Rd e−2πitξf1(t + x 2)f2(t − x 2) dt.
It is well-known that the Wigner distribution is the symplectic Fourier transform of the ambiguity function, see e.g., [21]. In other words, cf. [28, Lemma 4.3.4], (37) W (f1, f2)(x, ξ) = FUA(f1, f2)(x, ξ), f1, f2 ∈ L2(Rd),
where the operator U is the rotation UF (x, ξ) = F (ξ, −x) of a function F on R2d. We need the following norm equivalence.
Lemma 4.1. For s ∈ R, 1 ≤ p, q ≤ ∞, the following equivalence holds kW (f1, f2)kM1⊗vsp,q = kA(f1, f2)kW (FLp,Lq
vs)≍ kVf2f1kW (FLp,Lqvs).
Proof. Let us observe that the weight vs, s ∈ R, is symmetric in each coordinate: vs(x, ξ) = vs(x, −ξ) = vs(−x, ξ) = vs(−x, −ξ).
Using (37), the connection between modulation and Wiener amalgam spaces (15) and the symmetry of the weights vs we can write
kW (f1, f2)kM1⊗vsp,q = kFUA(f1, f2)kM1⊗vsp,q = kUA(f1, f2)kW (FLp,Lq vs) = kA(f1, f2)kW (FLp,Lq
vs). Now a simple change of variables in (36) let us write
A(f1, f2)(x, ξ) = eπixξVf2f1(x, ξ).
It was proved in [11, Proposition 3.2] that the function F (x, ξ) = eπixξ is in the Wiener amalgam space W (FL1, L∞). This means that, by the product properties for Wiener amalgam spaces, for every s ∈ R,
kA(f1, f2)kW (FLp,Lq
vs) .kF kW (FL1,L∞)kVf2f1kW (FLp,Lqvs) and since ¯F (x, ξ) = e−πixξ ∈ W (FL1, L∞) as well, with k ¯F k
W (FL1,L∞) = kF kW (FL1,L∞), we can analogously write
kVf2f1kW (FLp,Lq
vs) .kF kW (FL1,L∞)kA(f1, f2)kW (FLp,Lqvs). This proves the desired result.
This observations, together with the Wigner property W (f1, f2)(x, ξ) = W (f2, f1) let us translate Theorem 1.1 in terms of STFT acting from modulation spaces to Wiener amalgam spaces. Notice that the following two corollaries also hold for the ambiguity function A(f1, f2) in place of the STFT Vf1f2.
Corollary 4.1. Consider s ∈ R and assume that p1, p2, q1, q2, p, q ∈ [1, ∞] satisfy conditions (5) and (6). Then if f1 ∈ Mvp1|s|,q1(R
d) and f
2 ∈ Mvsp2,q2(Rd), we have Vf1f2 ∈ W (FLp, Lqvs)(R2d) with
(38) kVf1f2kW (FLp,Lq
vs)(R2d) .kf1kMvsp1,q1(Rd)kf2kMvsp2,q2(Rd). Viceversa, assume that there exists a constant C > 0 such that
(39) kVf1f2kW (FLp,Lq)(R2d) ≤ Ckf1kMp1,q1(Rd)kf2kMp2,q2(Rd), ∀f1, f2 ∈ S(R2d). Then (5) and (6) must hold.
The previous result has many special and interesting cases. Let us just give a flavour of the main important ones. For pi = qi, i = 1, 2, we obtain the following result.
Corollary 4.2. Assume that p1, p2, p, q ∈ [1, ∞] satisfy
(40) p1, p2 ≤ q, 1 p1 + 1 p2 ≥ 1 p + 1 q.
Then, for s ∈ R, if f1 ∈ Mvp1|s|(Rd) and f2 ∈ Mvsp2(Rd) we have Vf1f2 ∈ W (FLp, Lqvs)(R2d) with
(41) kVf1f2kW (FLp,Lq
vs)(R2d) .kf1kMv|s|p1 (Rd)kf2kMvsp2(Rd). Viceversa, assume that there exists a constant C > 0 such that
(42) kVf1f2kW (FLp,Lq)(R2d)≤ Ckf1kMp1(Rd)kf2kMp2(Rd), ∀f1, f2 ∈ S(R2d). Then (40) must hold.
Remark 4.3. The previous result holds also for the cross-Wigner distribution if we replace kVf1f2kW (FLp,Lvs)(Rq 2d) by kW (f1, f2)kMp,q
1⊗vs(R2d)
If we choose s = 0, p = q′ and q ≥ 2 in the previous result, we can refine some Lieb’s integral bounds for the ambiguity function showed in [36]. Namely, we obtain in particular the following sufficient conditions for boundedness.
Corollary 4.4. Assume q ≥ 2, p1, p2, q1, q2 ≤ q such that 1 p1 + 1 p2 ≥ 1 1 q1 + 1 q2 ≥ 1.
If fi ∈ Mpi,qi(Rd), i = 1, 2, then the ambiguity function satisfy A(f1, f2) ∈ Lq(R2d), with
kA(f1, f2)kLq(R2d) .kf1kMp1,q1(Rd)kf2kMp2,q2(Rd). Proof. Since FLq′
⊆ Lq for q ≥ 2, the inclusion relations for Wiener amalgam spaces give W (FLq′, Lq)(R2d) ⊆ W (Lq, Lq)(R2d) = Lq(R2d). The result then fol-lows by Corollary 4.1.
Observe that for pi = qi = 2, i = 1, 2, Mpi,qi(Rd) = L2(Rd) and we recapture Lieb’s bound, see [36, Theorem 1]. We also refer to [14, 15] for related estimates for the short-time Fourier transform.
5. Pseudodifferential operators
In this section we apply Theorem 1.1 to the study of pseudodifferential operators on modulation spaces. The key tool is the weak definition of a Weyl operator Lσ by means of a duality pairing between the symbol σ and the cross-Wigner distribution W (g, f ) as shown in (9).
The sharpest result concerning boundedness of pseudodifferential operators on (un-weighted) modulation spaces was proved by one of us with Tabacco and Wahlberg in [17, Theorem 1.1]. Such result covers previous sufficient boundedness conditions proved by Toft in [47, Theorem 4.3] and necessary boundedness conditions exhib-ited in our previous work [15, Proposition 5.3]. Our result in this framework ex-tends [17, Theorem 1.1] to weighted modulation spaces, thus widening the sufficient boundedness conditions presented by Toft in [48, Theorem 4.3]. Using Theorem 1.1 the proof of the following result is decidedly simple.
Theorem 5.1. Assume s ≥ 0, pi, qi, p, q ∈ [1, ∞], i = 1, 2, are such that
(43) min{ 1 p1 + 1 p′ 2 , 1 q1 + 1 q′ 2} ≥ 1 p′ + 1 q′. and (44) q ≤ min{p′1, q1′, p2, q2}.
Then the pseudodifferential operator T , from S(Rd) to S′(Rd), having symbol σ ∈ M1⊗vsp,q (R2d), extends uniquely to a bounded operator from Mp1,q1
vs (Rd) to Mvsp2,q2(Rd), with the estimate
(45) kT fkMp2,q2
vs .kσkM1⊗vsp,q kfkMvsp1,q1.
Vice-versa, if (45) holds for s = 0 and for every f ∈ S(Rd), σ ∈ S′(R2d), then (43) and (44) must be satisfied.
Proof. Assume σ ∈ M1⊗vsp,q (R2d), f ∈ Mvsp1,q1(Rd) such that (43) and (44) are satis-fied. For g ∈ Mp′2,q′2
v−s (Rd) Theorem 1.1 says that the cross-Wigner distribution is in M1⊗v−sp′,q′ (R2d), provided that p
1, q1, p′2, q2′ ≤ q′ and
min{1/p1+ 1/p′2, 1/q1+ 1/q1′} ≥ 1/p′ + 1/q′,
that are conditions (44) and (43), respectively. Thereby there exists a constant C > 0 such that |hσ, W (g, f)i| ≤ kakM1⊗vsp,q (R2d)kW (g, f)k M1⊗v−sp′,q′ (R2d) ≤ CkfkMvsp1,q1(Rd)kgkMp′2,q′2 v−s (Rd) .
Since |hLσf, gi| = |hσ, W (g, f)i|, this concludes the proof of the sufficient condi-tions. The necessary conditions are proved in [17, Theorem 1.1].
We now present sharp boundedness results for localization operators.
Let us mention that, since their introduction by Daubechies [20] as a mathemati-cal tool to lomathemati-calize a signal in the time-frequency plane, they have been investigated by many authors in the field of signal analysis, see [6, 13, 25, 40, 47, 51, 46] and references therein. Localization operators with Gaussian windows are well-known in quantum mechanics, under the name of anti-Wick operators [5, 41].
A localization operator Aϕ1,ϕ2
a with symbol a and windows ϕ1, ϕ2 is defined as (46) Aϕ1,ϕ2a f (t) =
Z
R2d
a(x, ξ)Vϕ1f (x, ξ)MξTxϕ2(t) dxdξ .
In signal analysis the meaning is as follows: first, analyse the signal f by taking the STFT Vϕ1f , then localize f by multiplying with the symbol a (if in particular a = χΩ, for some compact set Ω ⊆ R2d, it is considered only the part of f that lives on the set Ω in the time-frequency plane), then reconstruct the signal by superposition of time-frequency shifts with respect to the window ϕ2. If ϕ1(t) = ϕ2(t) = e−πt
2 , then Aa = Aϕ1,ϕ2a is the classical Anti-Wick operator and the mapping a → Aϕ1a ,ϕ2 is interpreted as a quantization rule [5, 41, 51].
Rewriting a localization operator Aϕ1,ϕ2
a as a Weyl operator, cf. (10), we can investigate boundedness properties for localization operators as boundedness con-ditions for Weyl operators having symbols a ∗ W (ϕ2, ϕ1), see (11). Again it comes into play Theorem 1.1.
Theorem 5.2. Assume s ≥ 0, the indices pi, qi, p, q ∈ [1, ∞], i = 1, 2, fulfil the relations (43) and (44). Consider r ∈ [1, 2]. If a ∈ M1/vp,q−s(R
2d) and ϕ
1, ϕ2 ∈ Mr
v2s(Rd), then the localization operator Aϕ1,ϕ2a is continuous from Mvsp1,q1(Rd) to Mp2,q2 vs (Rd) with kAϕ1,ϕ2a kop.kakMp,q 1/v−skϕ1kM r 2skϕ2kMv2sr .
Proof. Using Theorem 1.1 for ϕ1, ϕ2 ∈ Mv2sr (Rd) we obtain that W (ϕ2, ϕ1) ∈ M1⊗v2s1,∞ , for every r ∈ [1, 2]. Now the convolution relations in Proposition 2.1, in the form M1⊗v−sp,q ∗ M1⊗v2s1,∞ ⊆ M1⊗vp,qs, yield that the Weyl symbol σ = a ∗ W (ϕ2, ϕ1) belongs to M1⊗vsp,q . The result now follows from Theorem 5.1.
Remark 5.3. (i) The previous result extends Theorem 3.2 in [13] and Theorem 4.11 in [48] for this particular choice of weights. We observe that further exten-sions of Theorem 5.2 can be considered by using more general polynomial weights satisfying condition (4.17) in [48].
(ii) Using the same techniques as in the proof Theorem 5.2 one can study conditions on symbols and window functions such that the operator Aϕ1,ϕ2
a is in the Schatten class Sp, cf., e.g. [13, Theorem 3.4].
6. Further applications: The Cohen Class
The Cohen class (12) was introduced by Cohen in [8] essentially to circumvent the problem of the lack of positivity of the Wigner distribution. Many different kinds of kernels were proposed, in particular we recall for τ ∈ [0, 1] \ {1/2} the τ -kernels
στ(x, ξ) = 2d |2τ − 1|de
2πi2τ −12 xξ,
which provide the τ -Wigner distributions Wτ(f, f ) [7, Proposition 5.6]: Wτ(f, f ) = W (f, f ) ∗ στ.
We recall that such distributions can be used in the definition of the τ -pseudodifferential operators, see e.g., [7, 47]. Another important kernel is the Cohen kernel Θσ, which yields the Born-Jordan distribution [10, 11, 12], given by [12, Prop. 3.4]
Θσ(ζ1, ζ2) = (
−2 Ci(4π|ζ1ζ2|), (ζ1, ζ2) ∈ R2, d = 1
F(χ{|s|≥2}|s|d−2)(ζ1, ζ2), (ζ1ζ2) ∈ R2d, d ≥ 2,
where Ci(t) is the cosine integral function. It was shown in [12, Sec. 4] that στ, τ ∈ [0, 1] \ {1/2}, and Θσ belong to the modulation space M1,∞(R2d). Inspired by this result, Theorem 1.2 shows continuity properties for elements of the Cohen class having kernels in the modulation space M1,∞(R2d). Let us prove Theorem 1.2. The main ingredient will be Theorem 1.1.
Proof of Theorem 1.2. If f ∈ Mp1,q1
vs (Rd), with p1, q1 satisfying (13), Theorem 1.1 gives that the Wigner distribution is in the corresponding M1⊗vsp,q (R2d). Then the result follows by the inclusion relation M1⊗vp,qs ∗ M1,∞ ⊆ Mp,q
1⊗vs, s ≥ 0 (see Prop. 2.1).
Observe that the indices’ assumptions of Theorem 1.2 coincide with those of Corollary 3.4. Hence the continuity properties on modulation spaces of these Cohen kernels coincide with those of the Wigner distribution. In other words, the time-frequency properties of these Cohen distributions resemble those of the Wigner distribution.
Acknowledgements
The first author was partially supported by the Italian Local Project “Analisi di Fourier per equazioni alle derivate parziali ed operatori pseudo-differenziali”, funded by the University of Torino, 2013.
References
[1] A. Benedek and R. Panzone. The space Lp with mixed-norm. Duke Math. J., 28:301–324,
1961.
[2] A. B´enyi, K. Gr¨ochenig, K.A. Okoudjou and L.G. Rogers. Unimodular Fourier multipliers for modulation spaces. J. Funct. Anal., 246(2): 366-384, 2007.
[3] ´A. B´enyi and K. A. Okoudjou. Bilinear pseudodifferential operators on modulation spaces. J. Fourier Anal. Appl., 100 (3):301–313, 2004.
[4] A. B´enyi and K.A. Okoudjou. Time-frequency estimates for pseudodifferential operators. Contemporary Math., Amer. Math. Soc., 428:13–22, 2007.
[5] F. A. Berezin. Wick and anti-Wick symbols of operators. Mat. Sb. (N.S.), 86(128):578–610, 1971.
[6] P. Boggiatto, E. Cordero, and K. Gr¨ochenig. Generalized Anti-Wick operators with symbols in distributional Sobolev spaces. Integral Equations and Operator Theory, 48(4):427–442, 2004.
[7] P. Boggiatto, G. De Donno and A. Oliaro. Time-frequency representations of Wigner type and pseudo-differential operators. Trans. Amer. Math. Soc., 362(9):4955–4981, 2010. [8] L. Cohen. Generalized phase-space distribution functions. J. Math. Phys., 7:781–786, 1966. [9] L. Cohen. Time Frequency Analysis: Theory and Applications. Prentice Hall, 1995.
[10] E. Cordero, M. de Gosson and F. Nicola. On the Invertibility of Born-Jordan Quantization. J. Math. Pures Appl., 105(4):537–557, 2016.
[11] E. Cordero, M. de Gosson and F. Nicola. On the reduction of the interferences in the Born-Jordan distribution. Appl. Comput. Harmon. Anal., In Press. arXiv:1601.03719
[12] E. Cordero, M. de Gosson and F. Nicola. Time-frequency Analysis of Born-Jordan Pseudo-differential Operators. Submitted. arXiv:1601.05303.
[13] E. Cordero and K. Gr¨ochenig. Time-frequency analysis of localization operators. J. Funct. Anal., 205 (1):107–131, 2003.
[14] E. Cordero and F. Nicola. Sharp continuity results for the short-time Fourier transform and for localization operators. Monatsh. Math., 162:251–276, 2011.
[15] E. Cordero and F. Nicola. Pseudodifferential operators on Lp, Wiener amalgam and
modu-lation spaces. Int. Math. Res. Notices, 10:1860–1893, 2010.
[16] E. Cordero and K. Okoudjou. Multilinear localization operators. J. Math. Anal. Appl., 325 (2):1103-1116, 2007.
[17] E. Cordero, A. Tabacco and P. Wahlberg. Schr¨odinger-type propagators, pseudodifferential operators and modulation spaces. J. London Math. Soc., 88(2):375–395, 2013.
[18] E. Cordero, J. Toft and P. Wahlberg. Sharp results for the Weyl product on modulation spaces. J. Funct. Anal., 267(8):3016–3057, 2014.
[19] W. Czaja. Boundedness of pseudodifferential operators on modulation spaces. J. Math. Anal. Appl., 284(1):389–396, 2003.
[20] I. Daubechies. Time-frequency localization operators: a geometric phase space approach. IEEE Trans. Inform. Theory, 34(4):605–612, 1988.
[21] M. de Gosson. Symplectic methods in Harmonic Analysis and in Mathematical Physics, Birkh¨auser, 2011.
[22] H. G. Feichtinger. Banach convolution algebras of Wiener’s type, In Proc. Conf. “Function, Series, Operators”, Budapest August 1980, Colloq. Math. Soc. J´anos Bolyai, 35, 509–524, North-Holland, Amsterdam, 1983.
[23] H. G. Feichtinger. Banach spaces of distributions of Wiener’s type and interpolation. In Proc. Conf. Functional Analysis and Approximation, Oberwolfach August 1980, Internat. Ser. Numer. Math., 69:153–165. Birkh¨auser, Boston, 1981.
[24] H. G. Feichtinger. Generalized amalgams, with applications to Fourier transform. Canad. J. Math., 42(3):395–409, 1990.
[25] H. G. Feichtinger and K. Nowak. A First Survey of Gabor Multipliers. In H. G. Feichtinger and T. Strohmer, editors, Advances in Gabor Analysis. Birkh¨auser, Boston, 2002.
[26] Y. V. Galperin. Young’s Convolution Inequalities for weighted mixed (quasi-) norm spaces. J. Ineq. Spec. Funct., 5(1):1–12, 2014.
[27] L. Grafakos and R. H. Torres. Multilinear Calder´on-Zygmund theory, Adv. Math., 165(1): 124–164, 2002.
[28] K. Gr¨ochenig. Foundations of Time-Frequency Analysis. Birkh¨auser, Boston, 2001.
[29] K. Gr¨ochenig and C. Heil. Modulation spaces and pseudodifferential operators. Integral Equa-tions Operator Theory, 34:439–457, 1999.
[30] K. Gr¨ochenig and C. Heil. Counterexamples for boundedness of pseudodifferential operators. Osaka J. Math., 41:681–691, 2004.
[31] K. Gr¨ochenig and T. Strohmer. Pseudodifferential operators on locally compact abelian groups and Sj¨ostrand’s symbol class. J. Reine Angew. Math., 613:121–146, 2007.
[32] K. Johansson, S. Pilipovi´c, N. Teofanov and J. Toft. Sharp convolution and multiplication estimates in weighted spaces, Analysis and Applications, 13(5):457–480, 2015.
[33] M. Kobayashi and M. Sugimoto. The inclusion relation between Sobolev and modulation spaces. J. Funct. Anal., 260(11):3189–3208, 2011.
[34] D. Labate. Time-frequency analysis of pseudodifferential operators. Monatshefte Math., 133:143–156, 2001.
[35] D. Labate. Pseudodifferential operators on modulation spaces. J. Math. Anal. Appl., 262:242– 255, 2001.
[36] L. H. Lieb. Integral bounds for radar ambiguity functions and Wigner distributions. J. Math. Phys., 31, No.3, 1990.
[37] S. Molahajloo, K. A. Okoudjou and G. E. Pfander. Boundedness of Multilinear Pseudo-differential Operators on Modulation Spaces. J. Fourier Anal. Appl., 1–35, 2016. DOI:10.1007/s00041-016-9461-2
[38] S. Pilipovi´c, N. Teofanov and J. Toft. Micro-local analysis in Fourier-Lebesgue and modula-tion spaces. Part II, J. Pseudo-Differ. Oper. Appl., 1:341–376, 2010.
[39] S. Pilipovi´c, N. Teofanov and J. Toft. Micro-Local Analysis in Fourier-Lebesgue and modu-lation Spaces. Part I, J. Fourier Anal. Appl., 17:374–407, 2011.
[40] J. Ramanathan and P. Topiwala. Time-frequency localization via the Weyl correspondence. SIAM J. Math. Anal., 24(5):1378–1393, 1993.
[41] M. A. Shubin. Pseudodifferential Operators and Spectral Theory. Springer-Verlag, Berlin, second edition, 2001. Translated from the 1978 Russian original by Stig I. Andersson. [42] J. Sj¨ostrand. An algebra of pseudodifferential operators. Math. Res. Lett., 1(2):185–192, 1994. [43] T. Strohmer, Pseudodifferential operators and Banach algebras in mobile communications.
Appl. Comput. Harmon. Anal., 20(2):237–249, 2006.
[44] M. Sugimoto and N. Tomita. Boundedness properties of pseudo-differential operators and Calder`on-Zygmund operators on modulation spaces. J. Fourier Anal. Appl., 14(1):124–143, 2008.
[45] K. Tachizawa. The boundedness of pseudo-differential operators on modulation spaces. Math. Nachr., 168:263–277, 1994.
[46] N. Teofanov. Continuity and Schattenvon Neumann Properties for Localization Operators on Modulation Spaces. Mediterr. J. Math. 13:745–758, 2016.
[47] J. Toft. Continuity properties for modulation spaces, with applications to pseudo-differential calculus. I. J. Funct. Anal., 207(2):399–429, 2004.
[48] J. Toft. Continuity properties for modulation spaces, with applications to pseudo-differential calculus. II. Ann. Global Anal. Geom., 26(1):73–106, 2004.
[49] J. Ville. Th´eorie et Applications de la Notion de Signal Analytique, Cˆables et Transmission, 2:61–74, 1948.
[50] E. Wigner. On the Quantum Correction for Thermodynamic Equilibrium. Physical Review 40:749–759, 1932.
[51] M. W. Wong. Wavelets Transforms and Localization Operators, volume 136 of Operator Theory Advances and Applications. Birkhauser, 2002.
Dipartimento di Matematica, Universit`a di Torino, Dipartimento di Matematica, via Carlo Alberto 10, 10123 Torino, Italy
E-mail address: elena.cordero@unito.it
Dipartimento di Scienze Matematiche, Politecnico di Torino, corso Duca degli Abruzzi 24, 10129 Torino, Italy
E-mail address: fabio.nicola@polito.it
Dipartimento di Matematica, Universit`a di Torino, via Carlo Alberto 10, 10123 Torino, Italy
Dipartimento di Scienze Matematiche, Politecnico di Torino, corso Duca degli Abruzzi 24, 10129 Torino, Italy
E-mail address: elena.cordero@unito.it E-mail address: fabio.nicola@polito.it
